Solution of Nonlinear Elliptic Boundary Value Problems and Its Iterative Construction

We study a kind of nonlinear elliptic boundary value problems with generalized p-Laplacian operator. The unique solution is proved to be existing and the relationship between this solution and the zero point of a suitably defined nonlinear maximal monotone operator is investigated. Moreover, an iterative scheme is constructed to be strongly convergent to the unique solution. The work done in this paper is meaningful since it combines the knowledge of ranges for nonlinear operators, zero point of nonlinear operators, iterative schemes, and boundary value problems together. Some new techniques of constructing appropriate operators and decomposing the equations are employed, which extend and complement some of the previous work

Quantum phases of SrCu2(BO3)2 from high-pressure thermodynamics

We report heat capacity measurements of SrCu$_2$(BO$_3$)$_2$ under high pressure along with simulations of relevant quantum spin models and map out the $(P,T)$ phase diagram of the material. We find a first-order quantum phase transition between the low-pressure quantum dimer paramagnet and a phase with signatures of a plaquette-singlet state below T = $2$ K. At higher pressures, we observe a transition into a previously unknown antiferromagnetic state below $4$ K. Our findings can be explained within the two-dimensional Shastry-Sutherland quantum spin model supplemented by weak inter-layer couplings. The possibility to tune SrCu$_2$(BO$_3$)$_2$ between the plaquette-singlet and antiferromagnetic states opens opportunities for experimental tests of quantum field theories and lattice models involving fractionalized excitations, emergent symmetries, and gauge fluctuations.Comment: 6 pages + 8 pages supplemental informatio

The vital role of hole-carriers for superconductivity in pressurized black phosphorus

The influence of carrier type on superconductivity has been an important issue for understanding both conventional and unconventional superconductors [1-7]. For elements that superconduct, it is known that hole-carriers govern the superconductivity for transition and main group metals [8-10]. The role of hole-carriers in elements that are not normally conducting but can be converted to superconductors, however, remains unclear due to the lack of experimental data. Here we report the first in-situ high pressure Hall effect measurements on single crystal black phosphorus, measured up to ~ 50 GPa, and find a correlation between the Hall coefficient and the superconducting transition temperature (TC). Our results reveal that hole-carriers play a vital role in developing superconductivity and enhancing TC. Importantly, we also find a Lifshitz transition in the high-pressure cubic phase at ~17.2GPa, which uncovers the origin of a puzzling valley in the superconducting TC-pressure phase diagram. These results offer insight into the role of hole-carriers in developing superconductivity in simple semiconducting solids under pressure.Comment: 9 pages anf 3 figure